Liquid chromatography, and in particular high performance size exclusion chromatography, HPSEC, is a useful tool for the characterization of polymers. Typically, samples are prepared and injected into a chromatograph where they are pumped through columns that separate the molecules. In the case of HPSEC, molecules are separated by their hydrodynamic size; smaller molecules tend to remain longer in the interstices of the columns and therefore elute at later times than larger molecules. Historically, the chromatograph with its separating columns and concentration sensitive detector was calibrated by using nearly monodisperse polymeric standards spanning a broad range of molecular weight, MW, that bracketed those expected for the unknown samples being processed and analyzed. The MWs present in the unknown sample were thus derived from a measurement of the time required for each separated fraction of sample to pass through the chromatograph relative to the corresponding times for the narrow calibration standards. Since the mobile phase is generally pumped through the chromatograph at a constant rate, the time of passage of a sample through the system may be represented equivalently in terms of the cumulative volume of fluid eluted, i.e. the so-called elution volume.
Calibration-dependent techniques contain a number of sources of error, both random and systematic. Random errors are caused by variations in chromatographic conditions from run to run and differences in baseline and peak settings. Systematic errors arise from several causes. First, the calibration curve itself is generally only an approximation to the relationship between the logarithm of the MW and the elution volume. The MW values of the standards contain uncertainties, many having been obtained from other calibration curve-based measurements. Second, the conformational differences that may exist between the calibration standards and the unknowns, which will result in elution times at considerable variance with those expected to correspond to the unknown's true MW, will exacerbate further these systematic errors and make them quite difficult to detect. Finally, since some time interval inevitably elapses between the calibration and the measurement, drifts in system parameters such as flow rate and temperature become important.
Nevertheless, even if results generated from standards were inaccurate, the reproducibility of such measurements still provided an important basis for their utility. Indeed, as pointed out by Papazian and Murphy in their 1990 article in the Journal of Liquid Chromatography, repeated analyses of the same samples over many months and even years of measurements are used to estimate the relative uncertainties of the results presented. These uncertainties could be used to monitor the stability of the calibration procedures as well as the degradation of the columns themselves. But virtually every measurement of an HPSEC separation reported in the literature presents MWs without any estimate of the precision of the reported results. Software packages which process the collected data often report results to six or more significant figures with complete disregard for the fact that such accuracy is impossible.
With the advent of in-line light scattering detectors, the need to calibrate was no longer required, since a light scattering detector combined with a concentration detector permitted the determination of MWs and sizes, and their distributions, on an absolute basis. The intrinsic inaccuracies of results based on the calibration standards themselves, which were rarely of the same configuration as the unknown being analyzed, no longer affected the final results. This eliminates much of the systematic error discussed above, but the random errors still remain. In-line light scattering measurements, especially those including many simultaneous angles such as performed by the system of light scattering detectors manufactured by Wyatt Technology Corporation under the registered trade name DAWN, generally include so many data that one is naturally challenged to ask if the data are sufficient to calculate directly a measure of the precision of the results based on a single chromatographic run. Despite the fact that such amounts of collected data have been available for some time, it has heretofore gone unrecognized that estimates of precision were possible. During my study of many of these data collections, it has become apparent to me how these critical numbers could be generated.
Under normal circumstances, each detector in a chromatograph gives a signal which contains a peak, or several peaks, rising out of a flat baseline. In FIG. 1 appears the signal from one of the light scattering detectors along with the concentration-sensitive detector. In order to calculate a MW and rms radius for each slice, one must compute the ratio of the light scattering signal for each angle to the concentration signal, and fit these data to a model as a function of angle. When appropriate scaling is chosen, the value of the fit at zero angle, called the "intercept", is related to the MW, and the ratio of slope to intercept at zero angle is related to the mean square radius. These techniques have been thoroughly developed in the literature, for example in the articles by Zimm which appeared in the Journal of Chemical Physics beginning in 1948.
More specifically, consider the molecules, having been separated by the chromatograph, which elute at a particular volume corresponding to a slice i. These molecules are dissolved at a concentration c.sub.i which may be measured by the concentration sensitive detector. In addition, the molecules scatter light which can be measured by the light scattering detector at a set of angles .theta..sub.j. The light scattered from the molecules is described by excess Rayleigh ratios R(.theta..sub.j). At suitably low concentrations, these measured excess Rayleigh ratios are related to the weight average molecular weight M.sub.i, and the second virial coefficient A.sub.2 by the relation ##EQU1## Here, the quantity K* is an optical parameter defined by EQU K*4.pi..sup.2 n.sub.0.sup.2 (dn/dc).sup.2 .lambda..sub.0.sup.-4 N.sub.A.sup.-1 ( 2)
where n.sub.0 is the solvent refractive index, dn/dc is the specific refractive index increment of the solution, .lambda..sub.0 is the vacuum wavelength of the incident light, and N.sub.A is Avogadro's number. The quantity P.sub.i (.theta.) is called the scattering function or form factor for slice i and may always be written as a polynomial in sin.sup.2 (.theta./2): ##EQU2## where &lt;r.sup.2 &gt;.sub.i is the mean square radius of the scattering molecules averaged over the distribution present at slice i. If the molecules are known to be of a particular form, the expression for P.sub.i (.theta.) can be written in closed form. For example, if the molecules obey the random coil approximation, common for many types of polymer molecules, the scattering function can be written ##EQU3##
The calculations may also be performed using a reciprocal version of Eq. (1), namely ##EQU4## All the same analysis techniques described below apply to both Eqs. (1) and (6).
Light scattering detectors at very low and very high angles are sometimes unusable due to high noise levels from particulates or stray light. Even when usable, these detectors often have more noise than mid-range angles. This noise shows up in the baseline as well as in the peaks. The noise level of the concentration detector is typically, but not always, less than for the light scattering detectors.
Ideally, It would be very desirable to be able to calculate the uncertainty in the MW and rms radius for each slice from which comparable uncertainties for an entire peak could be generated using standard methods of error propagation. This would provide a statistical estimate of the typical variation one would expect to see from repeated measurements of the same sample. Such a calculation could save considerable time as well as provide an independent measure of the precision of the measurement. Unfortunately, measurement of the uncertainties associated with a single slice would require that repeated measurements be made as each slice elutes. Such measurements would have to be made over long enough periods of time to permit the collection of the plurality of values from which their corresponding standard deviations could be calculated, following standard statistical procedures. Yet the fundamental concept of making a chromatographic separation relies upon the requirement that the eluting sample be divided into slices, each one of which occurs over a time frame very short compared to that required for the whole sample. Measurement of a single slice cannot be frozen in time in order to quantify its fluctuation.
Despite these apparent contradictions, my invention shows clearly how the precision of determinations using chromatographic separations may now be estimated quite easily. Although the method has focused almost entirely upon analyses of data collected in the preferred embodiment of an in-line multi-angle light scattering detector, my invention may certainly be practiced by those skilled in the art for other types of light scattering detector systems incorporating one, two, three, or more angles for their measurements, as well as for measurements and precision determinations of viscometric quantities.